Question

How do I find the trigonometric form of the complex number $3 - 4i?$

Answer 1

As we know that a complex number is a number which can be expressed in the $a + bi$ form, where $a$ and $b$ are real numbers and $i$ is the imaginary number. It means it consists of both real and imaginary parts. Now any complex number can be converted into trigonometric form also known as polar form i.e. $z = \left| z \right|\left( {\cos \theta + i\sin \theta } \right)$. First we will calculate the squares of the real part and then the square of the imaginary part and then we will add the above calculated values and find the square root to get the absolute value of the given complex number.

Complete step by step solution:
As per the given we have a complex number $z = 3 - 4i$.
The real part of the given complex number is $3$ and the imaginary part of the above complex number is $- 4$.
We will now calculate the square of both the real and the imaginary part, the square of the real part of the given complex number is ${3^2} = 9$.
Square of the imaginary part of the given complex number is ${( - 4)^2} = 16$.
Sum of the squares of the real and imaginary part of the complex number is $9 + 16 = 25$. Now we will find the square root of the sum of the squares of the real and the imaginary part: $\sqrt {25} = 5$.
Now the polar form is $z = \left| z \right|\left( {\cos \theta + i\sin \theta } \right)$, where $\cos \theta = \dfrac{a}{{\left| z \right|}}$ and $\sin \theta = \dfrac{b}{{\left| z \right|}}$.
So here $\cos \theta = \dfrac{3}{5} = 0.6$ and $\sin \theta = \dfrac{{ - 4}}{5} = - 0.8$.
The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion i.e. $\theta = \arcsin ( - 0.8) = - 0.93rad$, since the $\theta = \arctan \dfrac{{ - 4}}{3}$ and the inverse of this produces an angle in the fourth quadrant.
Now putting the values in the trigonometric form we have $z = 5\left[ {\cos \left( { - 0.93} \right) + i\sin \left( { - 0.93} \right)} \right]$.

Hence trigonometric form is $z = 5\left[ {\cos \left( { - 0.93} \right) + i\sin \left( { - 0.93} \right)} \right]$.

Note: We should be careful while calculating the values and in the square of the imaginary part we should note that the square of any negative number is always positive, the negative sign changes. We can also directly calculate the absolute value of the complex number $z = 3 - 4i$ which is also denoted by $\left| z \right|$, by the formula $\left| {a + bi} \right| = \sqrt {{a^2} + {b^2}}$, where $a$ is the real part and $b$ is the imaginary part.